Global stability of mutualistic interactions among three species population model with continuous time delay

A. B. Munde; M. B. Dhakne

doi:10.26637/mjm101/010

Abstract

This paper deals with the study on a mathematical model consisting of mutualistic interactions among three species with continuous time delay. The delay kernels are being convex combinations of suitable nonnegative and normalized functions, the linear chain trick gives an expanded system of ordinary differential equations with the same stability properties as the original integro-differential system. Global stability is discussed by constructing Lyapunov function. It has been shown that equilibrium state of the model is globally stable. Finally, numerical simulations supporting our theoretical results are also included.

Keywords:

Mutualism model, local and global stabilities, Lyapunov function, population dynamics, time delay

Mathematics Subject Classification:

92B05, 92D25, 93A30, 93C15, 93D99

Authors Imprint References Funding Related Articles Downloads

A. B. Munde Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad-431 004, India. https://orcid.org/0000-0003-2213-1987
M. B. Dhakne Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad-431 004, India.

Pages: 98-105
Date Published: 01-01-2013
Vol. 1 No. 01 (2013): Malaya Journal of Matematik (MJM)

E. Beretta and Y. Takeuchi, Global stability of single species diffusion Volterra models continuous time delays, Bulletin of Mathematical Biology, 49(4)(1987), 431-448. DOI: https://doi.org/10.1016/S0092-8240(87)80005-8

A.W. Busekros, Global stability in ecological systems with continuous time delay, SIAM J. Appl. Math., 35(1)(1978), 123-134. DOI: https://doi.org/10.1137/0135011

J.M. Cushing, Integro-differential equations and delay models in population dynamics, Lecture-Notes in Biomathematics, No. 20, Springer-Verlag, Berlin, 1977. DOI: https://doi.org/10.1007/978-3-642-93073-7

C.H. Feng and P.H. Chao, Global stability for the Lotka Volterra mutualistic system with time delay, Tunghai Science, 8(2006), 81-107.

K. Gopalsamy, Stability on the oscillations in delay differential equations of population dynamics, Academic Press, New York, 1993. DOI: https://doi.org/10.1007/978-94-015-7920-9

K. Gopalsamy, Global asymptotic stability in Volterra’s population systems, J. Math. Biology, 19(1984), 157-168. DOI: https://doi.org/10.1007/BF00277744

J.K. Hale and P. Waltman, Persistence in finite-dimensional systems, SIAM J. Math. Anal., 20(1989), 388-395. DOI: https://doi.org/10.1137/0520025

S.W. Harlan, The effect of time lags on the stability of the equilibrium state of a population growth equation, J. Math. Biology, 5(1978), pp. 115-120. DOI: https://doi.org/10.1007/BF00275894

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, San Diego, 1993.

N. Mc Donald, Time lags in biological models, Lecture Notes in Biomathematics, No. 27, Springer-Verlag, Berlin 1978. DOI: https://doi.org/10.1007/978-3-642-93107-9

D. Mukherjee, Permanence and global attractivity for facultative mutualism system with delay, Math. Meth. Appl. Sci., 26(2003), 1-9. DOI: https://doi.org/10.1002/mma.275

V.P. Shukla, Condition for global stability of two species population models with discrete time delay, Bull. Math. Biology, 45(5)(1983), 793-805. DOI: https://doi.org/10.1016/S0092-8240(83)80026-3

F. Solimano and E. Beretta, Existence of a globally asymptotically stable equilibrium in Volterra models with continuous time delay, J. Math. Biology, 18(1983), 93-102. DOI: https://doi.org/10.1007/BF00280659

Y. Xia, Existence of positive solutions of mutualism systems with several delays, Advances in Dynamical Systems and Applications, 1(2)(2006), 209-217.